Is the following sequence of partial sums bounded?
$$\sum_{n=1}^{N}{e^{i\,n!\,x}}$$
where $x$ is in $\left(0,2\pi\right)$ and $x$ is not a rational multiple of $\pi$.
2026-04-24 19:52:51.1777060371
Sum of complex exponential
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Hint: what happens if $x$ is a rational multiple of $\pi$?
EDIT: The set of $x$ such that the sequence is unbounded is a dense $G_\delta$, so it also contains uncountably many $x$ such that $x/\pi$ is irrational.